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Symmetric equilibria for a beam with a nonlinear elastic foundation. (English) Zbl 0815.34014
Using a standard variational technique due to P. Rabinowitz, the authors study the following boundary value problem which connects with the elastic beam theory: $$u^{(4)} (x) + g(x,u(x)) = 0$$, $$u''(0) = u''(1) = 0$$, $$u'''(0) = - f(u(0))$$, $$u'''(1) = f(u(1))$$. Under some natural restrictions on $$f$$ and $$g$$, having mechanical nature, they establish existence of solutions to the problem in $$H^ 2_ s(0,1)$$, a subspace of symmetrical functions $$u \in H^ 2_ s(0,1)$$, $$u(x) = u(1-x)$$.

MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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